I am trying to learn more about Ricci scalar curvature. I am trying to get an image in my head of what scalar curvature actually represents about the curvature of a manifold. The most familiar image I have of a scalar giving me information about a space is a scalar field, like temperature. Is that the right way to think about scalar curvature? If not, how should one envision scalar curvature?
According to Wikipedia here, scalar curvature can be characterized as a multiple of the average sectional curvatures at a point.
For 2D manifolds the scalar curvature is the Gauss curvature. Positive means the manifold closes in on itself, negative means it spreads out like a saddle.
Some of this parallel remains in higher dimensions, since the sign of curvature determines the asymptotic volume comparison for geodesic balls (positive curvature $\implies$ less-than -Euclidean volume, negative $\implies$ greater than Euclidean). However this is on small scale only, and therein lies a problem: the scalar curvature does not deliver much global geometric information.
Indeed, the asymptotic volume definition implies that scalar curvature is additive under products. Thus, by taking product with a sufficiently scaled-down compact surface of constant negative curvature (think of a tiny double torus), we can make scalar curvature negative, as long as it was bounded to begin with. And by taking product with a tiny sphere, the scalar curvature can be made positive.
These examples explain why there isn't a picture of what a manifold of positive/negative scalar curvature looks like: it can look like pretty much anything. (There are some obstructions to manifolds admitting positive scalar curvature, but they are not easy to visualize.)